Rainbow Vertex-Connection and Forbidden Subgraphs

• Main Authors: Li Wenjing, Li Xueliang, Zhang Jingshu
• Format: Article
• Language: English
• Published: Sciendo 2018-02-01
• Series: Discussiones Mathematicae Graph Theory
• Subjects: vertex-rainbow path; rainbow vertex-connection; forbidden sub-graphs; 05c15; 05c35; 05c38; 05c40
• Online Access: https://doi.org/10.7151/dmgt.2004

A path in a vertex-colored graph is called vertex-rainbow if its internal vertices have pairwise distinct colors. A vertex-colored graph G is rainbow vertex-connected if for any two distinct vertices of G, there is a vertex-rainbow path connecting them. For a connected graph G, the rainbow vertex-connection number of G, denoted by rvc(G), is defined as the minimum number of colors that are required to make G rainbow vertex-connected. In this paper, we find all the families ℱ of connected graphs with |ℱ| ∈ {1, 2}, for which there is a constant kℱ such that, for every connected ℱ-free graph G, rvc(G) ≤ diam(G) + kℱ, where diam(G) is the diameter of G.